Covariant integral quantisation using coherent states for semi-direct product groups is implemented for the motion of a particle on the circle. In this case, the phase space is the cylinder, which is viewed as a left coset of the Euclidean group E(2). Coherent states issued from fiducial vectors are labeled by points in the cylinder and depend also on extra parameters. We carry out the corresponding quantisations of the basic classical observables, particularly the angular momentum and the 2π-periodic discontinuous angle function. We compute their corresponding lower symbols. The quantum localisation on the circle is examined through the properties of the angle operator yielded by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.
The question how to quantize a classical system where an angle $\ensuremath{\varphi}$ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace $\ensuremath{\varphi}$ by the smooth periodic functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$ and $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$. In the case of the canonical pair $(\ensuremath{\varphi},{p}_{\ensuremath{\varphi}})$, where ${p}_{\ensuremath{\varphi}}$ is the orbital angular momentum (OAM), the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}={\ensuremath{\varphi}∊\mathbb{R}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}2\ensuremath{\pi},{p}_{\ensuremath{\varphi}}∊\mathbb{R}}$ has the global topological structure ${S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$ of a cylinder on which the Poisson brackets of the three functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi},\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$, and ${p}_{\ensuremath{\varphi}}$ obey the Lie algebra of the Euclidean group
E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group
E(2) or of its covering groups. A crucial point is that, due to the fact that the subgroup $\mathrm{SO}(2)\ensuremath{\cong}{S}^{1}$ is multiply connected, these representations allow for fractional OAM $l=\ensuremath{\hbar}(n+\ensuremath{\delta}),n∊\mathbb{Z},\ensuremath{\delta}∊[0,1)$. Such $\ensuremath{\delta}\ensuremath{\ne}0$ have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}$ in terms of unitary representations of
E(2) allows for two types of ``coherent'' states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.